Invertible means one-to-one and onto. In particular, we only say that a map $f:A \to B$ is invertible if there is another map $g:B \to A$ such that both $f \circ g$ and $g \circ f$ are the identity maps over their respective spaces.
Of course, any one-to-one map can be made invertible by restricting the codomain to the image of the map. Similarly (assuming the axiom of choice), we can make any onto map invertible by restricting the domain to an appropriate subset.
In some contexts, it makes sense to call the (natural) logarithm the inverse of the exponential map, even when this restriction of the domain is not explicitly stated. As you might expect, the domain of the logarithm must be the positive numbers.